Frequently Used Equations

Mechanics

velocity	acceleration	equations of motion	newton's 2nd law
v =  Δs Δt	a =  Δv Δt	v = v0 + at	∑ F = m a
		x = x0 + v0t + ½at2	∑ F =  dp dt
v =  ds dt	a =  dv dt	v2 = v02 + 2a(x − x0)
		v = ½(v + v0)
weight	centrip. accel.	momentum	efficiency
W = m g	ac =  v2 r	p = m v	ℰ =  Wout Ein
dry friction
ƒ = μN	ac = − ω2 r
impulse	impulse–momentum	work	work–energy
J = F Δt	F Δt = m Δv	W = FΔs cos θ	FΔs cos θ = ΔE
J =  ⌠ ⌡ F dt	⌠ ⌡ F dt = Δp	W =  ⌠ ⌡ F · ds	⌠ ⌡ F · ds = ΔE
kinetic energy	potential energy	power
K = ½mv2	ΔU = −  ⌠ ⌡ F · ds	P =  ΔW Δt	P =  dW dt
gravitational p.e.
ΔUg = mgΔh	F = − ∇U	P = Fv cos θ	P = F · v
angular velocity	angular acceleration	equations of rotation	2nd law for rotation
ω =  Δθ Δt	α =  Δω Δt	ω = ω0 + αt	∑ τ = I α
		θ = θ0 + ω0t + ½αt2	∑ τ =  dL dt
ω =  dθ dt	α =  dω dt	ω2 = ω02 + 2α(θ − θ0)
		ω = ½(ω + ω0)
v = ω × r	a = α × r − ω2 r
torque	moment of inertia	rotational work	rotational power
τ = rF sin θ	I = ∑ mr2	W = τΔθ	P = τω cos θ
τ = r × F	I =  ⌠ ⌡  r2 dm	W =  ⌠ ⌡  τ · dθ	P = τ · ω
rotational k.e.	angular momentum	frequency	hooke's law
K = ½Iω2	L = mrv sin θ	ƒ =  1 T	F = − k Δx
	L = r × p		elastic p.e.
	L = I ω	ω = 2πƒ	Us = ½kΔx2
universal gravitation	gravitational field	gravitational p.e.	gravitational potential
Fg = −  Gm1m2  r̂ r2	g = −  Gm  r̂ r2	Ug = −  Gm1m2 r	Vg = −  Gm r
orbital speed	escape speed	s.h.o.	simple pendulum
v = √  Gm r	v = √  2Gm r	T = 2π √  m k	T = 2π √  ℓ g
density	pressure	pressure in a fluid	buoyancy
ρ =  m V	P =  F A	P = P0 + ρgh	B = ρgVdisplaced
volume flow rate	mass flow rate	volume continuity	mass continuity
φ =  V t	I =  m t	A1v1 = A2v2	ρ1A1v1 = ρ2A2v2
bernoulli's equation		aerodynamic drag	kinematic viscosity
P1 + ρgy1 + ½ρv12 = P2 + ρgy2 + ½ρv12		R = ½ρCAv2	ν =  η ρ
dynamic viscosity	reynolds number	mach number	froude number
η =  F/A Δvx/Δz	Re =  ρvD η	Ma =  v c	Fr =  v √gℓ
η =  F/A dvx/dz
young's modulus	shear modulus	bulk modulus	surface tension
F  = Y  Δℓ A ℓ0	F  = S  Δx A y	F  = B  ΔV A V0	γ =  F ℓ

Thermal Physics

solid expansion	liquid expansion	sensible heat	mean kinetic energy
Δℓ = αℓ0ΔT	ΔV = βV0ΔT	Q = mcΔT	〈K〉 = 3/2kT
ΔA = 2αA0ΔT	ideal gas law	latent heat	internal energy
ΔV = 3αV0ΔT	PV = nRT	Q = mL	ΔU = 3/2nRΔT
	PV = NkT		ΔU = 3/2NkΔT
r.m.s. speed	heat flow rate	thermal conduction	stefan-boltzmann law
vrms = √  3kT μ	Φ =  ΔQ Δt	Φ =  kAΔT ℓ	Φ = εσA(T4 − T04)
			wien displacement law
vrms = √  3RT M	Φ =  dQ dt		λmax =  b T
efficiency	c.o.p.	thermodynamic work	entropy
ℰreal = 1 −  QC QH	COPreal =  QC QH − QC	W = − ⌠ ⌡ P dV	ΔS =  ΔQ T
ℰideal = 1 −  TC TH	COPideal =  TC TH − TC	first law	S = k log w
		ΔU = Q + W

Waves & Optics

periodic waves	frequency	beat frequency	intensity
v = ƒλ	ƒ =  1 T	fbeat = fhigh − flow	I =  〈P〉 A
power level		interference fringes
β = 10 log  I  = 20 log  P I0 P0		nλ = d sin θ
		nλ  ≈  x d L
index of refraction	snell's law	mirrors and lenses	mirrors only
n =  c v	n1 sin θ1 = n2 sin θ2	1  =  1  +  1 ƒ do di	ƒ =  r 2
	critical angle
	sin θc =  n2 n1	M =  hi  =  di ho do

Electricity & Magnetism

coulomb's law	electric field
F = k  q1q2 r2	E =  Fe q	E = k ∑  q  r̂ r2	E = k  ⌠ ⌡ dq  r̂ r2
field and potential	electric potential
E = −  ∆V d	ΔV =  ΔUE q	V = k ∑  q r	V = k  ⌠ ⌡ dq r
E = − ∇V
capacitance	plate capacitor	cylindrical capacitor	spherical capacitor
C =  Q V	C =  κε0A d	C =  2πκε0ℓ ln (b/a)	C =  4πκε0 (1/a) − (1/b)
capacitive p.e.
U =  1  CV2 =  1   Q2  =  1  QV 2 2 C 2
electric current	ohm's law	electric resistance	electric power
I =  Δq Δt I =  dq dt	V = IR	R =  ρℓ A	P = VI = I2R =  V2 R
	E = ρ J
resistors in series	resistors in parallel	capacitors in series	capacitors in parallel
Rs = ∑ Ri	1  = ∑  1 Rp Ri	1  = ∑  1 Cs Ci	Cp = ∑ Ci
magnetic force
FB = qvB sin θ	FB = q v × B	FB = IℓB sin θ	dFB = I dℓ × B
biot-savart law	solenoid	straight wire	parallel wires
B =  μ0I ⌠ ⌡ ds × r̂ 4π r2	B = µ0nI	B =  μ0I 2πr	FB  =  μ0   I1 I2 ℓ 2π r
motional emf	electric flux	magnetic flux	induced emf
V = Bℓv	ΦE = EA cos θ	ΦB = BA cos θ	ℰ = −  ΔΦB Δt ℰ = −  dΦB dt
	ΦE =  ⌠ ⌡ E · dA	ΦB =  ⌠ ⌡ B · dA
gauss's law	no one's law	faraday's law	ampere's law
∯  E · dA =  Q ε0	∯ B · dA =   0	∮E · ds = −  dΦB dt	∮B · ds = μ0ε0  dΦE  + μ0I dt
∇ · E =  ρ ε0	∇ · B =   0	∇ × E = −  ∂B ∂t	∇ × B = μ0ε0  ∂E  + μ0 J ∂t

Modern Physics

time dilation	length contraction	relativistic mass	relative velocity
t' =  t √(1 − v2/c2)	ℓ' = ℓ √(1 − v2/c2)	m' =  m √(1 − v2/c2)	u' =  u + v 1 + uv/c2
relativistic energy	relativistic momentum	energy-momentum	mass-energy
E =  mc2 √(1 − v2/c2)	p =  mv √(1 − v2/c2)	E2 = p2c2 + (mc2)2	E = mc2
relativistic doppler effect	photon energy	photoelectric effect	photon momentum
λ0  =  ƒ  = √ ⎛ ⎝ 1 − v/c ⎞ ⎠ λ ƒ0 1 + v/c	E = hf	Kmax = E − ϕ = h(ƒ − ƒ0)	p =  h λ
schroedinger's equation		uncertainty principle	rydberg equation
iℏ  ∂  Ψ(r, t) = −  ℏ2  ∇2Ψ(r, t) + V(r)Ψ(r, t) ∂t 2m		Δpx Δx ≥  ℏ  2	1  = −R∞  ⎛ ⎝ 1  −  1 ⎞ ⎠ λ n2 n02
Eψ(r) = −  ℏ2  ∇2ψ(r) + V(r)ψ(r) 2m		ΔE Δt ≥  ℏ  2	half life
			N = N02−t/τ